Subgroups — #36524639

Order 22784.

666 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26029187	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19151676	C₂	1	1	yes	maximal, characteristic
3	#26019143	C₂	1	4	yes	maximal, complemented
4	#26019096	C₂	1	4	yes	maximal, complemented
5	#26019023	C₂	1	1	yes	maximal, characteristic, complemented
6	#26018677	C₂	1	1	yes	maximal, characteristic
7	#26018655	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018536	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3101	C₂	1	1	yes	maximal, characteristic
10	C₂²×#13083558	—	2	1	no	—
11	#19151729	—	2	1	no	—
12	#19151729	C₂²	1	2	yes	complemented
13	Q₈×#10623494	—	2	1	no	—
14	Q₈×#10623494	C₂²	1	2	yes	complemented
15	#19151676	—	2	1	no	—
16	#19151676	—	2	1	no	—
17	#19151676	C₂²	1	2	yes	—
18	#19151676	C₂²	1	2	yes	—
19	#19151651	C₂²	1	1	yes	characteristic
20	#19151625	—	2	1	no	—
21	#19151622	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊#10623494	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊#10623494	—	2	1	no	—
24	#19151554	—	2	1	no	—
25	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
26	#19151349	C₂²	1	4	yes	complemented
27	Q₈⋊#10623494	C₂²	1	4	yes	complemented
28	D₄⋊#10623494	C₂²	1	4	yes	—
29	#19151320	C₂²	1	2	yes	complemented
30	C₈×#10623494	C₂²	1	2	yes	complemented
31	#533×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
33	#483×C₈₉	C₄	1	2	yes	—
34	#476×C₈₉	C₄	1	2	yes	—
35	#461×C₈₉	C₄	1	2	yes	—
36	#450×C₈₉	C₄	1	2	yes	—
37	C₄×SD₁₆×C₈₉	C₂²	1	4	yes	—
38	#431×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#13083558	—	2	2	no	—
40	C₂×#13083558	—	2	2	no	—
41	C₂×#13083558	—	2	1	no	—
42	C₂×#13083558	—	4	4	no	—
43	C₂²×#10623494	—	2	1	no	—
44	C₂²×#10623494	D₄	1	1	yes	characteristic
45	C₂²×#10623494	D₄	1	1	yes	characteristic
46	C₄⋊#10623494	—	2	2	no	—
47	C₄⋊#10623494	—	2	2	no	—
48	C₄⋊#10623494	—	2	1	no	—
49	C₄⋊#10623494	—	2	1	no	—
50	C₄⋊#10623494	—	2	1	no	—
51	C₄⋊#10623494	C₂³	1	2	yes	—
52	C₄⋊#10623494	C₂³	1	2	yes	—
53	C₂²⋊#10623494	—	2	2	no	—
54	C₂²⋊#10623494	—	2	2	no	—
55	C₂²⋊#10623494	—	2	1	no	—
56	#15667923	—	2	2	no	—
57	C₄×#10623494	—	2	2	no	—
58	C₄×#10623494	D₄	1	1	yes	characteristic
59	C₄×#10623494	D₄	1	1	yes	characteristic
60	C₄×#10623494	C₂³	1	2	yes	—
61	C₄×#10623494	C₂³	1	1	yes	characteristic
62	C₄×#10623494	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₈₉	C₂×C₄	1	1	yes	characteristic
64	C₂³×C₄×C₈₉	—	2	1	no	—
65	C₂×SD₁₆×C₈₉	C₂×C₄	1	4	yes	—
66	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
67	C₄⋊Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
68	#125×C₈₉	—	2	1	no	—
69	C₂²⋊Q₈×C₈₉	—	2	1	no	—
70	C₂²⋊Q₈×C₈₉	C₂×C₄	1	2	yes	—
71	C₄×Q₈×C₈₉	—	2	1	no	—
72	C₄×Q₈×C₈₉	C₂³	1	2	yes	—
73	C₄×D₄×C₈₉	—	2	1	no	—
74	C₄×D₄×C₈₉	—	2	1	no	—
75	C₄×D₄×C₈₉	C₂×C₄	1	2	yes	—
76	C₄×D₄×C₈₉	C₂³	1	2	yes	—
77	#117×C₈₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
81	#106×C₈₉	C₂×C₄	1	4	yes	—
82	C₄⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
83	#103×C₈₉	C₂×C₄	1	4	yes	—
84	#102×C₈₉	C₂×C₄	1	4	yes	—
85	C₂²⋊C₈×C₈₉	C₂×C₄	1	2	yes	—
86	C₄×C₈×C₈₉	C₂³	1	2	yes	—
87	#13083558	—	4	2	no	—
88	#13083558	—	4	4	no	—
89	#13083558	—	4	4	no	—
90	C₂×#10623494	—	2	2	no	—
91	C₂×#10623494	—	2	2	no	—
92	C₂×#10623494	—	2	2	no	—
93	C₂×#10623494	—	2	2	no	—
94	C₂×#10623494	—	4	2	no	—
95	C₂×#10623494	SD₁₆	1	4	yes	complemented
96	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
97	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
98	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
99	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
100	C₂⁴×C₈₉	—	2	1	no	—
101	C₂×Q₈×C₈₉	—	2	1	no	—
102	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
103	C₂×D₄×C₈₉	—	2	1	no	—
104	C₂×D₄×C₈₉	—	4	2	no	—
105	C₂×D₄×C₈₉	C₂²×C₄	1	2	yes	—
106	C₂²×C₄×C₈₉	—	2	1	no	—
107	C₂²×C₄×C₈₉	—	2	2	no	—
108	C₂²×C₄×C₈₉	—	2	2	no	—
109	C₂²×C₄×C₈₉	—	4	2	no	—
110	C₂²×C₄×C₈₉	—	4	2	no	—
111	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
112	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
114	SD₁₆×C₈₉	—	4	4	no	—
115	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
116	C₄⋊C₄×C₈₉	—	2	1	no	—
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
123	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
124	C₂²⋊C₄×C₈₉	—	2	2	no	—
125	C₂²⋊C₄×C₈₉	—	2	2	no	—
126	C₂²⋊C₄×C₈₉	—	2	2	no	—
127	C₄²×C₈₉	—	2	2	no	—
128	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₄²×C₈₉	#41	1	1	yes	characteristic
131	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂³×C₈₉	—	2	2	no	—
133	C₂³×C₈₉	—	4	2	no	—
134	C₂³×C₈₉	—	4	2	no	—
135	C₂³×C₈₉	—	4	2	no	—
136	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
137	Q₈×C₈₉	—	2	4	no	—
138	Q₈×C₈₉	—	4	2	no	—
139	D₄×C₈₉	—	2	2	no	—
140	D₄×C₈₉	—	2	2	no	—
141	D₄×C₈₉	—	4	2	no	—
142	C₂×C₄×C₈₉	—	2	2	no	—
143	C₂×C₄×C₈₉	—	2	2	no	—
144	C₂×C₄×C₈₉	—	2	2	no	—
145	C₂×C₄×C₈₉	—	2	2	no	—
146	C₂×C₄×C₈₉	—	4	1	no	—
147	C₂×C₄×C₈₉	—	4	2	no	—
148	C₂×C₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	—	4	2	no	—
150	C₂×C₄×C₈₉	—	4	4	no	—
151	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
152	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
153	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
154	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
155	C₂×C₄×C₈₉	#135	1	2	yes	—
156	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
157	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
159	C₈×C₈₉	—	2	4	no	—
160	#10623494	—	2	1	no	—
161	#10623494	—	2	1	no	—
162	#10623494	—	4	2	no	—
163	#10623494	—	4	2	no	—
164	#10623494	—	4	2	no	—
165	#10623494	C₂×SD₁₆	1	2	yes	—
166	#10623494	C₂×SD₁₆	1	2	yes	—
167	C₂²×C₈₉	—	2	1	no	—
168	C₂²×C₈₉	—	2	1	no	—
169	C₂²×C₈₉	—	4	2	no	—
170	C₂²×C₈₉	—	4	2	no	—
171	C₂²×C₈₉	—	4	2	no	—
172	C₂²×C₈₉	—	8	4	no	—
173	C₂²×C₈₉	C₄×SD₁₆	1	2	yes	—
174	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
175	C₄×C₈₉	—	2	1	no	—
176	C₄×C₈₉	—	2	1	no	—
177	C₄×C₈₉	—	2	1	no	—
178	C₄×C₈₉	—	4	2	no	—
179	C₄×C₈₉	—	4	2	no	—
180	C₄×C₈₉	—	4	2	no	—
181	C₄×C₈₉	C₄×SD₁₆	1	2	yes	—
182	C₄×C₈₉	#568	1	2	yes	—
183	C₄×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
184	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
185	#17930	—	89	1	no	maximal, Sylow
186	C₂×C₈₉	—	2	2	no	—
187	C₂×C₈₉	—	8	2	no	—
188	C₂×C₈₉	#2930	1	1	yes	characteristic
189	C₂×C₈₉	C₂×C₄×SD₁₆	1	1	yes	characteristic
190	C₂×C₈₉	#2955	1	1	yes	characteristic
191	#3101	D₈₉	1	1	yes	characteristic
192	#2960	—	89	1	no	—
193	C₂×#436	—	89	1	no	—
194	#1590	—	89	4	no	—
195	#1583	—	89	4	no	—
196	#1570	—	89	1	no	—
197	#1486	—	89	1	no	—
198	#1481	—	89	1	no	—
199	#1450	—	89	1	no	—
200	C₈₉	#17930	1	1	yes	characteristic, complemented, Sylow
201	C₂²×M₁₆	—	178	1	no	—
202	#533	C₂×D₈₉	1	1	yes	characteristic
203	C₂×C₄×D₄	C₂×D₈₉	1	1	yes	characteristic
204	#483	Dic₈₉	1	2	yes	—
205	#476	Dic₈₉	1	2	yes	—
206	#461	Dic₈₉	1	2	yes	—
207	#450	Dic₈₉	1	2	yes	—
208	#446	—	89	2	no	—
209	#446	—	178	1	no	—
210	C₈×Q₈	—	89	2	no	—
211	C₈×Q₈	—	178	1	no	—
212	C₄×SD₁₆	C₂×D₈₉	1	4	yes	—
213	#436	—	89	2	no	—
214	#436	—	89	2	no	—
215	#436	—	178	1	no	—
216	#436	—	178	1	no	—
217	#431	—	89	1	no	—
218	#431	C₂×D₈₉	1	1	yes	characteristic
219	#424	—	178	1	no	—
220	#423	—	89	1	no	—
221	C₂×C₄⋊C₈	—	89	1	no	—
222	C₂×C₂²⋊C₈	—	178	1	no	—
223	#405	—	178	1	no	—
224	C₂×C₄×C₈	—	89	1	no	—
225	#334	—	89	4	no	—
226	Q₈⋊C₈	—	89	4	no	—
227	D₄⋊C₈	—	89	4	no	—
228	#322	—	89	2	no	—
229	C₈²	—	89	2	no	—
230	C₂²×D₄	C₂×Dic₈₉	1	1	yes	characteristic
231	C₂³×C₄	—	2	1	no	—
232	C₂×SD₁₆	C₂×Dic₈₉	1	4	yes	—
233	C₂×M₁₆	—	356	4	no	—
234	C₂×M₁₆	—	178	2	no	—
235	C₂×M₁₆	—	178	2	no	—
236	C₂×M₁₆	—	178	1	no	—
237	C₂²×C₈	—	89	1	no	—
238	C₂²×C₈	—	89	1	no	—
239	C₂²×C₈	—	178	1	no	—
240	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
241	C₄⋊Q₈	C₂×Dic₈₉	1	1	yes	characteristic
242	#125	—	2	1	no	—
243	C₂²⋊Q₈	—	2	1	no	—
244	C₂²⋊Q₈	C₂×Dic₈₉	1	2	yes	—
245	C₄×Q₈	—	2	1	no	—
246	C₄×Q₈	C₂²×D₈₉	1	2	yes	—
247	C₄×D₄	—	2	1	no	—
248	C₄×D₄	—	2	1	no	—
249	C₄×D₄	C₂×Dic₈₉	1	2	yes	—
250	C₄×D₄	C₂²×D₈₉	1	2	yes	—
251	#117	—	2	1	no	—
252	C₂×C₄⋊C₄	C₂×Dic₈₉	1	1	yes	characteristic
253	C₂×C₂²⋊C₄	—	2	1	no	—
254	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
255	#106	C₂×Dic₈₉	1	4	yes	—
256	C₄⋊C₈	—	89	2	no	—
257	C₄⋊C₈	—	89	2	no	—
258	C₄⋊C₈	—	178	2	no	—
259	C₄⋊C₈	—	178	2	no	—
260	C₄⋊C₈	—	178	1	no	—
261	C₄⋊C₈	—	178	1	no	—
262	C₄⋊C₈	—	178	1	no	—
263	C₄⋊C₈	C₂×Dic₈₉	1	2	yes	—
264	#103	C₂×Dic₈₉	1	4	yes	—
265	#102	C₂×Dic₈₉	1	4	yes	—
266	C₂²⋊C₈	—	178	2	no	—
267	C₂²⋊C₈	—	178	2	no	—
268	C₂²⋊C₈	—	178	1	no	—
269	C₂²⋊C₈	C₂×Dic₈₉	1	2	yes	—
270	#97	—	178	2	no	—
271	C₄×C₈	—	89	1	no	—
272	C₄×C₈	—	89	1	no	—
273	C₄×C₈	—	89	1	no	—
274	C₄×C₈	—	89	1	no	—
275	C₄×C₈	—	89	2	no	—
276	C₄×C₈	—	178	2	no	—
277	C₄×C₈	C₂²×D₈₉	1	2	yes	—
278	C₂⁴	—	2	1	no	—
279	C₂×Q₈	—	2	1	no	—
280	C₂×Q₈	C₂²×Dic₈₉	1	2	yes	—
281	C₂×D₄	—	2	1	no	—
282	C₂×D₄	—	4	2	no	—
283	C₂×D₄	C₂²×Dic₈₉	1	2	yes	—
284	C₂²×C₄	—	2	1	no	—
285	C₂²×C₄	—	2	2	no	—
286	C₂²×C₄	—	2	2	no	—
287	C₂²×C₄	—	4	2	no	—
288	C₂²×C₄	—	4	2	no	—
289	C₂²×C₄	C₂²×Dic₈₉	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
291	C₂²×C₄	#13083573	1	1	yes	characteristic
292	SD₁₆	—	4	4	no	—
293	M₁₆	—	356	4	no	—
294	M₁₆	—	356	4	no	—
295	M₁₆	—	356	2	no	—
296	C₂×C₈	—	89	4	no	—
297	C₂×C₈	—	89	1	no	—
298	C₂×C₈	—	89	1	no	—
299	C₂×C₈	—	89	1	no	—
300	C₂×C₈	—	89	1	no	—
301	C₂×C₈	—	356	2	no	—
302	C₂×C₈	—	178	2	no	—
303	C₂×C₈	—	178	2	no	—
304	C₂×C₈	—	178	2	no	—
305	C₂×C₈	—	178	2	no	—
306	C₂×C₈	C₂²×Dic₈₉	1	4	yes	—
307	C₄⋊C₄	—	2	1	no	—
308	C₄⋊C₄	—	2	1	no	—
309	C₄⋊C₄	—	2	1	no	—
310	C₄⋊C₄	—	2	1	no	—
311	C₄⋊C₄	—	2	2	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
314	C₄⋊C₄	C₂²×Dic₈₉	1	2	yes	—
315	C₂²⋊C₄	—	2	2	no	—
316	C₂²⋊C₄	—	2	2	no	—
317	C₂²⋊C₄	—	2	2	no	—
318	C₄²	—	2	2	no	—
319	C₄²	C₂²×Dic₈₉	1	1	yes	characteristic
320	C₄²	D₄×D₈₉	1	1	yes	characteristic
321	C₄²	#13083573	1	1	yes	characteristic
322	C₄²	C₂³×D₈₉	1	1	yes	characteristic
323	C₂³	—	2	2	no	—
324	C₂³	—	4	2	no	—
325	C₂³	—	4	2	no	—
326	C₂³	—	4	2	no	—
327	C₂³	D₄×Dic₈₉	1	1	yes	characteristic
328	Q₈	—	2	4	no	—
329	Q₈	—	4	2	no	—
330	D₄	—	2	2	no	—
331	D₄	—	2	2	no	—
332	D₄	—	4	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	2	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	4	1	no	—
338	C₂×C₄	—	4	2	no	—
339	C₂×C₄	—	4	4	no	—
340	C₂×C₄	—	4	2	no	—
341	C₂×C₄	—	4	2	no	—
342	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
343	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
344	C₂×C₄	D₄×Dic₈₉	1	1	yes	characteristic
345	C₂×C₄	SD₁₆×D₈₉	1	2	yes	—
346	C₂×C₄	#15668033	1	2	yes	—
347	C₂×C₄	C₂³×Dic₈₉	1	1	yes	characteristic
348	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
349	C₂×C₄	C₂×#13083573	1	1	yes	characteristic
350	C₈	—	2	4	no	—
351	C₈	—	89	2	no	—
352	C₈	—	89	2	no	—
353	C₈	—	356	2	no	—
354	C₈	—	356	2	no	—
355	C₈	—	356	2	no	—
356	C₈	—	178	1	no	—
357	C₈	—	178	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	4	2	no	—
362	C₂²	—	4	2	no	—
363	C₂²	—	8	4	no	—
364	C₂²	SD₁₆×Dic₈₉	1	2	yes	—
365	C₂²	C₂×D₄×Dic₈₉	1	1	yes	characteristic
366	C₄	—	2	1	no	—
367	C₄	—	2	1	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	2	no	—
370	C₄	—	4	2	no	—
371	C₄	—	4	2	no	—
372	C₄	SD₁₆×Dic₈₉	1	2	yes	—
373	C₄	#19152365	1	2	yes	—
374	C₄	C₂×SD₁₆×D₈₉	1	1	yes	characteristic
375	C₄	C₂×#15668033	1	1	yes	characteristic
376	C₂	—	2	2	no	—
377	C₂	—	8	2	no	—
378	C₂	#26028956	1	1	yes	characteristic
379	C₂	C₂×SD₁₆×Dic₈₉	1	1	yes	characteristic
380	C₂	#26029155	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	C₈₉×#3101	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₄×C₈₉	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	#3101	D₈₉
hypocenter	C₈₉	#17930
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3101

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #26029187	C₂	semidirect
3. #26019143	C₂	semidirect
4. #26019096	C₂	semidirect
5. #26019023	C₂	semidirect
7. #26018655	C₂	semidirect
8. #26018536	C₂	semidirect
12. #19151729	C₂²	semidirect
14. Q₈×#10623494	C₂²	semidirect
26. #19151349	C₂²	semidirect
27. Q₈⋊#10623494	C₂²	semidirect
29. #19151320	C₂²	semidirect
30. C₈×#10623494	C₂²	semidirect
95. C₂×#10623494	SD₁₆	semidirect
200. C₈₉	#17930	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#26029187	2
C₂×#19151676	2
#26019143	2
#26019096	2
#26019023	2
#26018677	2
#26018655	2
#26018536	2
C₈₉×#3101	2
#17930	89